# Question concerning math in relation to nature

1. Jan 9, 2004

### sero

I like to study and read about physics in my spare time. One thing that has always puzled me is how the scientists, particulary physicists and mathematicians can come up with a mathmatical equation for an event in nature.One thing I've tried to study to get a better understanding is the Lorenzt Transformation equation. I dont understand when to use division or multiplication,square roots or some more advanced mathmatics like calc plus anything past calc. I took up to calculus in college, but I am still baffled as how it would be used to model or explain a real world event.

1. are there any finite rules for when to use a particular type of mathmatics, or when to use one type with another? for example, how would I know when to use a square root or division ECT ?

Last edited: Jan 9, 2004
2. Jan 9, 2004

### mighty2000

It all starts with a question that you have to ask yourself. ie. You would use a square root to stay within a certain domain. If your experiment dictated that your solution would prove true with a certain range or domain, then you would use it in your equations. If you are measuring the volume of objects then you would use cubes and cube roots. Now I know this isn't true for all equations, but an example. Developing linear equations requires constants and cooeficients. These have to be measured and placed in the equations. Original equations should be based on the scientific method. Your original hypothesis will be tested over and over again with multiple formulas until a theory is developed.

I guess this might not be what you were looking for, but it is the best I could do considering your question.

Math is Math, either you know it or you don't.
Thats the way I look at it.

M2k

3. Jan 10, 2004

### pnaj

Another way to think about this in terms of building models for real life situations. This generally involves trying to relate empirical data (gathered by whatever means) in mathematical relationships that will also predict new data, that can be verified or 'honed' by further experiments. These models may be more or less accurate, but might still serve a purpose, or hint at deeper relationships, etc.

I think the simplest example is proportionality.

For an obvious example just to show the process, you might suppose that the time taken to complete a journey is related to the distance travelled, but what is the relationship?

Obviously, we start with the simplest journey - a straight line - and do experiments. We notice that, for the results to mean anything, we try to 'control' all factors that might affect the experiment, other than the two we are wanting to relate. That means, for example, keeping the motion constant, staying on a flat plane, etc. for all the trials.

After gathering together the data, we notice, empirically, that the distance travelled is equal to some constant number * the time taken.
So, d = k * t , where k is this constant that appeared in the data and ee can now predict how far we'll go for any particular time.
In physical terms, this states that t is directly proportional to d, and k is the constant of proportionality.

To go on with this ...

Someone might then ask what this k actually represents, in the real world, if anything. If we isolate K in the equation, we get k = d/t ... distance over time ... which is what we call (average) speed!

But, in general, this is only really useful over short distances and times, because most things don't even approximate to straight lines for very long.

Calculus was a major step forward in the calculation of motion ... you should really have a crack at it if you want to understand anything to do with motion.

Tell us what you do know about calculus and maybe we can push you on a bit.

4. Jan 10, 2004

### Integral

Staff Emeritus
The key to modeling the real world world is to think about what is changing and how it changes.

For example you can say that the change in a population is porptional to the number of individuals present at any time

$$\frac {\Delta p} {\Delta t} = kp$$

This leads to the standard simple population model.

The goal of modeling is to come up with a DIFFERENTIAL equation. To do this you need a good understanding of the physics which govern you system. If you have that understanding is is ofter pretty straight forward to express the differential equation. Once you have the differential equation you must be able to solve it.

Last edited: Jan 10, 2004
5. Jan 11, 2004

### HallsofIvy

Staff Emeritus
I really wondering if a person who starts off by saying "I dont understand when to use division or multiplication" is going to understand any of this. "When to use division and multiplication", i.e. what division and multiplication mean is one of the things you are supposed to learn in elementary school.

6. Jan 12, 2004

### sero

obviously, I guess I need to be more clear. I'm looking for some general rules of when to use these types when trying to take a natual event and express it mathmatically. I do know what each mean, did learn them in elementary school, but I wanted to ask in a setting where people that use what im asking about would understand the question and be able to anwser it so that it would help me get a better feel for it. not just division and multiplication, but more complex things like square roots.

I hope that makes it more clear so that I can get more constructive anwsers then the previous one.

7. Jan 12, 2004

### Staff: Mentor

At its most basic, this is the human ability for pattern recognition. Do an experiment, gather some data, and attach a pattern to the results.

The simplest examples come from Newtonian physics. There were some long held beliefs about motion that no one until Galileo did much experimenting on. For example, most people before Galileo believed heavier objects fall faster than lighter objects. A simple demonstration proved that wrong (Tycho Brahe, I think, used to demonstrate this by dropping food on the floor at parties). That's just one piece of the puzzle of course, but an important one - Newton is the one who put all the pieces together, recognized all the patterns, and formulated the math to describe it.
Sorry, that still isn't very clear: what do you mean by "when?" What operations you use are whatever operations that allow you to find the pattern in the data. Often you do this by plain, ordinary trial and error.

example:
1:10
2:20
3:30
4:40
5:50
See the pattern? It should be obvious: every number in the second colum is 10x the number in the first. In equation form, v=10t. Guess what - that's acceleration (gravitational, to be specific)!

1:5
2:20
3:45
4:80
5:125
See the pattern? This one is tougher. Knowing the first example though may help find the pattern in this one.

Now Newton probably just derived the second one from the first, nevertheless having real data to check is essential.

This is also why math and physics are so closely related.

Last edited: Jan 12, 2004
8. Jan 12, 2004